[UAI] Mixed DBN propagation

From: Todd Stephenson (todd@idiap.ch)
Date: Tue Nov 27 2001 - 10:23:42 PST

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    I am working with dynamic Bayesian networks with a mixture
    of continuous and discrete variables. I am using the
    propagation algorithm given in Lauritzen and Jensen (2001),
    "Stable local computation with conditional Gaussian distributions,"
    Statistics and Computing.

    Consider the following DBN, with discrete variables D[1] . . . D[N]
    and continous variables C[1] . . . C[N], where any variable can be
    unobserved:

    D[1] ---> D[2] ---> D[3] ---> D[4] ---> . . . ---> D[N]
     | | | | |
     | | | | |
     | | | | |
     | | | | |
     V V V V V
    C[1] ---> C[2] ---> C[3] ---> C[4] ---> . . . ---> C[N]

    Now, in order to have a decomposable graph after triangulation,
    there can not be any path between two non-adjacent discrete variables
    that passes only through continuous vertices (Proposition 7.9
    of Cowell et al., (1999) "Probabilistic Networks and Expert Systems).
    In this case, this means that all the discrete vertices D[1] . . . D[N]
    must be fully connected (because, since there is such a path between
    every possible pair of discrete vertices, the only solution is to
    connect them all with each other in triangulation).

    ============================================================

    My question is the following:

    Is there a method to avoid having to fully connect all these
    discrete variables and thus have to have a large clique containing
    all of the discrete variables? I would prefer, rather, to have
    smaller cliques (e.g. the clique {D[t],D[t+1],C[t],C[t+1]}) when
    doing propagation.

    Thanks in advance for any input.

    Todd 

    --
Todd Stephenson
Dalle Molle Institute for
      Perceptual Artificial Intelligence
PO Box 592
CH-1920 Martigny
Switzerland
Tel: +41 27 721 77 52
Fax: +41 27 721 77 12
Email:  Todd.Stephenson@idiap.ch
WWW: http://www.idiap.ch

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